Limit Cycles Bifurcations for a Class of 3-Dimensional Quadratic Systems

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• 作者：Chaoxiong Du (1)
  Qinlong Wang (2)
  Yirong Liu (3)
  
  1. Department of Mathematics ; Hunan Shaoyang University ; Shaoyang ; Hunan ; 422000 ; P.R. China
  2. Department of Mathematics ; Hezhou University ; Hezhou ; Guangxi ; 542800 ; P.R. China
  3. School of Mathematics ; Central South University ; Changsha ; Hunan ; 410075 ; P.R. China
  
• 关键词：3 ; Dimensional quadratic system ; Hopf bifurcation ; Singular values ; Limit聽cycles
• 刊名：Acta Applicandae Mathematicae
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：136
• 期：1
• 页码：1-18
• 全文大小：590 KB
• 参考文献：1. Heinz Kienitz, K.H. (2006) On the implementation of the eigenvalue method for limit cycle determination in nonlinear systems. Nonlinear Dyn. 45: pp. 25-30 CrossRef
  2. Du, C., Liu, Y., Chen, H. (2010) The bifurcation of limit cycles in Z n -equivariant vector fields. Appl. Math. Comput. 217: pp. 2041-2056 CrossRef
  3. Haiyan, Hu. (2009) New Development of Theory and Applications on Nonlinear Dynamics. Science, Beijing
  4. Llibre, J., Makhlouf, A., Badi, S. (2009) 3-dimensional Hopf bifurcation via averaging theory of second order. Discret. Continous Dyn. Syst. 25: pp. 1287-1295 CrossRef
  5. Shen, J., Chen, S., Lin, K. (2011) study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discret. Continous Dyn. Syst. 15: pp. 231-254 CrossRef
  6. Cardin, P.T., Carvalho, T., Llibre, J. (2012) Bifurcation of limit cycles from an n-dimensional linear center inside a class of piecewise linear differential systems. Nonlinear Anal., Theory Methods Appl. 75: pp. 143-152 CrossRef
  7. Barreira, L., Llibre, J., Valls, C. (2012) Bifurcation of limit cycles from a 4-dimensional center in R m in resonance 1: N. J. Math. Anal. Appl. 389: pp. 754-768 CrossRef
  8. Liu, Y. (2001) Theory of center-focus for a class of higher-degree critical points and infinite points. Sci. China Ser. A 44: pp. 37-48
  9. Liu, Y., Li, J. (1990) Theory of values of singular point in complex autonomous differential system. Sci. China Ser. A 33: pp. 10-24
  10. Eneral, V.F., Mahdi, A., Romanovski, V.G., Shafer, D.S. (2012) The center problem on a center manifold in R 3. Nonlinear Anal., Theory Methods Appl. 75: pp. 2614-2622 CrossRef
  11. Mahdi, A., Romanovski, V.G., Shafer, D.S. (2013) Stability and periodic oscillations in the Moon-Rand systems. Nonlinear Anal., Real World Appl. 14: pp. 294-313 CrossRef
  12. Kuznetsov, Y.A. (2004) Elements of Applied Bifurcation Theory. Springer, New York CrossRef
  13. Kuznetsov, Y.A. (1999) Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE鈥檚. SIAM J. Numer. Anal. 36: pp. 1104-1124 CrossRef
  14. Sotomayor, J., Mello, L.F. (2007) Denis de Carvalho Braga. Bifurcation analysis of the Watt governor system. Comput. Appl. Math. 26: pp. 19-44 CrossRef
  15. Du, C., Wang, Q., Huang, W. (2014) Three-dimensional Hopf bifurcation for a class of cubic Kolmogorov model. Int. J. Bifurc. Chaos 24: pp. 1-11
  16. Wang, Q., Liu, Y., Chen, H. (2010) Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems. Bull. Sci. Math. 134: pp. 786-798 CrossRef
  17. Andronov, A.A. (1973) Theory of Bifurcations of Dynamic Systems on a Plane. Halsted, New聽York/Toronto
  18. Chow, S.N., Hale, J.K. (1982) Methods of Bifurcation Theory. Springer, Berlin
  
• 刊物主题：Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9036

文摘

Our work is concerned with the Hopf bifurcation for a class of 3-dimensional quadratic symmetric vector field. By making use of singular values methods and computing carefully, we give the expressions of the first four focal values of the two singular point that system has. Both singular symmetric points can become fine foci of fourth order at the same time. Moreover, we obtain that from each one can bifurcate 4 small limit cycles under a certain coefficients鈥?perturbed condition, consequently, at most eight limit cycles can appear by simultaneous Hopf bifurcation.